Handlebody decompositions of three-manifolds and polycontinuous patterns

We introduce the concept of a handlebody decomposition of a three-manifold, a generalization of a Heegaard splitting, or a trisection. We show that two handlebody decompositions of a closed orientable three-manifold are stably equivalent. As an application to materials science, we consider a mathematical model of polycontinuous patterns and discuss a topological study of microphase separation of a block copolymer melt.


Introduction
A Heegaard splitting is a decomposition of a closed orientable three-manifold into two handlebodies of the same genus. It is well known that every closed orientable three-manifold admits a Heegaard splitting. By the Reidemeister-Singer theorem [1,2], two Heegaard splittings of a given three-manifold are stably equivalent, i.e. isotopic after a finite number of stabilizations.  Many generalizations of Heegaard splittings have been investigated. Gómez-Larrañaga [3] studied orientable three-manifolds decomposed into three solid tori. Coffey and Rubinstein analysed orientable three-manifolds built from three π 1 -injective handlebodies [4]. In [5], Koenig considered a trisection of a closed orientable three-manifold, which is an embedded branched surface decomposing the manifold into three handlebodies with connected pairwise intersections. Koenig introduced the notion of stabilization for a trisection and showed an analogue of the Reidemeister-Singer theorem for trisections of three-manifolds.
In this paper, we consider a generalization of all of the above. We define a handlebody decomposition to be a decomposition of a closed orientable three-manifold into a finite number of handlebodies (see definition 2.1 for the detailed definition). We will also introduce stabilizations for handlebody decompositions and show an analogue of the Reidemeister-Singer theorem for handlebody decompositions (see theorem 3.5).
The primary motivation of this study comes from materials science. We are interested in the characterization of bicontinuous patterns, tricontinuous patterns and polycontinuous patterns of microphase separation of a block copolymer melt (see §6b). See [6,7] for related research. A mathematical model of a bicontinuous (resp. tricontinuous or polycontinuous) pattern is a triply periodic non-compact surface (resp. tribranched surface or polyhedron) embedded in R 3 that divides it into two (resp. three or a finite number of) possibly disconnected submanifolds as shown in figure 1 (see definition 5.8 for more details). We are particularly interested in the case where the submanifolds are the open neighbourhood of networks.
If a bicontinuous pattern is triply periodic, then by considering the quotient of the action, the pattern induces a Heegaard splitting of the three-dimensional torus T 3 (see remark 7.1). If a polycontinuous pattern is triply periodic and satisfies suitable conditions, then it corresponds to a handlebody decomposition of T 3 (corollary 5.12). Hence a characterization of handlebody decompositions of T 3 gives that of triply periodic polycontinuous patterns. The Reidemeister-Singer-type theorem of polycontinuous patterns (corollary 6.3) follows from that of handlebody decompositions of T 3 . This point of view allows us to explain how two polycontinuous patterns are related, which will be discussed in §6b.
This paper is organized as follows. In §2, we define a handlebody decomposition of a three-manifold. In §3, we introduce several types of stabilization operations of handlebody decompositions and prove an analogue of the Reidemeister-Singer theorem for them. In §4, we particularly focus on decompositions of three-manifolds into three handlebodies. In §5, we study a mathematical model of polycontinuous patterns. We define polycontinuous patterns and, more generally, net-like patterns. The correspondence between triply periodic net-like patterns and handlebody decompositions of T 3 is given. In §6, we discuss stabilizations of net-like patterns. We also present how this research relates to the subject of materials science. In §7, we give characterizations of net-like patterns.

Handlebody decompositions of three-manifolds
We work in the piecewise linear category throughout this paper.
By a two-dimensional polyhedron P, we mean the underlying space of a non-collapsible locally finite two-dimensional complex such that the link of each vertex contains no isolated vertices. A connected component of the set of points of P having neighbourhoods homeomorphic to discs is called a sector. The set of all points not contained in the sectors is called its singular graph. A twodimensional polyhedron P is said to be simple if, after giving a structure of a complex in a suitable way, the link of each point in P is homeomorphic to one of the three models shown in figure 2. A point whose link is homeomorphic to the model in figure 2c is called a vertex of its singular graph. See Matveev [8] for more details.

Definition 2.1 (Handlebody decomposition).
Let M be a closed, connected, orientable threemanifold and P a connected compact two-dimensional polyhedron embedded in M. We call (H 1 , H 2 , . . . , H n ; P) a type-(g 1 , g 2 , . . . , g n ) handlebody decomposition of M if M\P = n i=1 H i , where H i is the interior of a handlebody of genus g i . The polyhedron P is called a partition for the decomposition. A handlebody decomposition is said to be proper if there is no simple closed curve in M\B that intersects a sector of P transversely once, where B is the singular graph of P. A handlebody decomposition is said to be simple if its partition is a simple polyhedron.

Remark 2.2.
(1) Let (H 1 , H 2 , . . . , H n ; P) be a type-(g 1 , g 2 , . . . , g n ) handlebody decomposition of M, and W i a handlebody of genus g i for i = 1, 2, . . . , n. Then there exists a continuous map ι i : W i → M such that the restriction of ι i to the interior of W i is a homeomorphism to H i . Then we have ι i (W i ) ∩ P = ι i (∂W i ) ∩ P. Suppose that the handlebody decomposition is proper. Then for the closure F of each sector, there exists a pair of handlebodies (W i , W j ) (i = j) such that F ⊂ ι i (∂W i ) ∩ ι j (∂W j ). We denote the union of all such surfaces F by F ij . (Note that F ij = F ji .) (2) In general, ι i may not be injective on the singular graph of P. If the decomposition is simple and proper, then ι i is a homeomorphism.
The notion of handlebody decompositions generalizes both Heegaard splittings [9] and trisections [5] of closed orientable three-manifolds. In fact, a simple proper handlebody decomposition with n = 2 is nothing but a Heegaard splitting, while that with n = 3, where each F ij is connected, is a trisection. By [10], any closed, connected, three-manifold M admits a simple (non-proper) type-(0) handlebody decomposition. Therefore, it is easily seen that for any sequence (g 1 , . . . , g n ) of non-negative integers, there exists a simple (possibly non-proper) type-(g 1 , . . . , g n ) handlebody decomposition of M.

Stable equivalence
This section discusses the stable equivalence of simple proper handlebody decompositions of a three-manifold. We assume that a handlebody decomposition is simple and proper
(0) Take a properly embedded arc α in H i , and an arc β in ∂H i such that the endpoints of α lie in the interior of F ij , and α is parallel to β in H i relative to the endpoints, i.e. the endpoints of α are equal to that of β, and α ∪ β bounds a disc in H i . Then we get a type-(g 1 , . . . , g n ) handlebody decomposition (H 1 , . . . , H n ) of M with where N(α) and int(N(α)) are a regular neighbourhood of α and its interior in H i , respectively. We call this operation a type-0 stabilization (along α). Conversely, we assume that there exist properly embedded discs D j of H j and E in H i such that the boundary of D j is in F ij , and the boundary of D j intersects that of E transversely exactly one point. Then we can perform the inverse operation of a type-0 stabilization. We call this operation a type-0 destabilization (along D j ). See figure 3a. (1) Take a properly embedded arc α on F jk such that the endpoints of α lie in the boundary of H i for i = j, k. Then we get a type-(g 1 , . . . , g n ) handlebody decomposition (H 1 , . . . , H n ) of M with We call this operation a type-1 stabilization (along α). Conversely, if there exists a nonseparating disc D i of H i such that the boundary of D i intersects the singular graph of the partition P transversely exactly two points, then we can perform the inverse operation of a type-1 stabilization. We call this operation a type-1 destabilization (along D i ). See figure 3b. (2) Take two points on the interior of F ij and that of F ik for j = k, and we connect the points by a properly embedded arc α in H i . Let β be an arc in ∂H i such that α is parallel to β. Then we get a type-(g 1 , . . . , g n ) handlebody decomposition (H 1 , . . . , H n ) of M with We call this operation a type-2 stabilization (along α). Conversely, if there exists a disc component D jk of F jk whose boundary intersects a properly embedded non-separating disc in H i transversely once, then we can perform the inverse operation of a type-2 stabilization. We call this operation a type-2 destabilization (along D jk  Figure 3. Stabilizations and destabilizations. Red curves represent the singular graphs. Both ends of the arc α of a type-0 stabilization are contained in F ij , whereas one end of the arc α of a type-2 stabilization is contained in F ij and the other is in F ik with j = k. A type-0 stabilization connects two parts of H j by the 1-handle N(α). On the other hand, a new branch locus and a new component D jk of F jk appear after a type-2 stabilization. (a) Type-0, (b) type-1 and (c) type-2. (Online version in colour.) Remark 3.2. Consider a type-(g 1 , g 2 , . . . , g n ) handlebody decomposition of a closed, connected, orientable three-manifold M with 3 ≤ n. For every g i ≤ g i , we can obtain a type-(g 1 , g 2 , . . . , g n ) handlebody decomposition of M by performing type-1 stabilizations repeatedly in a suitable way.

Definition 3.3.
A handlebody decomposition is said to be stabilized if it is obtained from another handlebody decomposition by a stabilization.
When n = 2, a type-0 stabilization is nothing but a stabilization of Heegaard splittings. In electronic supplementary material, we discuss the independence of these stabilizations.

(b) Stable equivalence theorem
This subsection will generalize Koenig's argument [5] on the stable equivalence of decompositions. We first recall the following operations on simple polyhedra embedded in a closed orientable three-manifold introduced by Matveev [8] and Piergallini [11] under our setting. (1) Let α be a properly embedded arc in F jk . A modification of P in a neighbourhood of α, as in figure 4a, is called a 0-2 move along α. By this operation, the number of vertices of P increases by two, and a new disc component appears in F il . Conversely, we can perform the inverse operation of 0-2 move along a disc component, D, of F il . We call the operation a 2-0 move along D. By this operation, the number of vertices of P decreases by two, and the disc component is removed from F il . (2) Let α be an edge of the singular graph of P. A modification of P in a neighbourhood of α, as in figure 4b, is called a 2-3 move along α. By this operation, the number of vertices of P increases by one, and a new disc component appears in F im . Conversely, we can perform the inverse operation of 2-3 move along a disc component, D, of F im . We call the operation a 3-2 move along D. By this operation, the number of vertices of P decreases by one, and the disc component is removed from F im .
We note that the above moves do not change the topological type of each handlebody of a decomposition.   Proof. Set F ij = H i ∩ H j and F ij = H i ∩ H j as in remark 2.2. We will prove the theorem in the following steps.
Step 0. In the case of n ≥ 4, we perform 0-2, 2-0, 2-3 moves and type-1 stabilization appropriately until it holds F ij = ∅ for any 3 ≤ i < j ≤ n. Then P becomes a simple polyhedron without vertices.
Step 3. We cover H 1 along S 1 with H 3 by type-1 stabilizations. Then it holds that H 3 = H 3 after handle slides.
Then it holds that H i = H i after handle slides.
If n = 3, after performing the operations described in the first half of Step 0, the decompositions H and H become trisections. Then, they are equivalent by using Koenig's theorem. Hence, in this proof, we assume that n ≥ 4.
Step 0. Put J = {(i, j) | 3 ≤ i < j ≤ n, F ij = ∅}. Let (i, j) be the minimum element of J in the lexicographical order. First, we change F ij to be connected if it is disconnected as follows. Take an arc properly embedded in the closure of ∂H i \ F ij that connects different components of F ij . If the arc is contained in some F ik , a type-1 stabilization along the arc decreases the number of components by one. Otherwise, we can perform a type-1 stabilization after 0-2 moves along the arc to decrease the number of components. Hence, by repeatedly applying this process finitely many times, we may assume that F ij is connected.
Next, take mutually disjoint arcs properly embedded in F ij so that they cut open F ij into a disc. We perform either a type-1 stabilization or a 0-2 move along each of the arcs according to whether both ends of the arc lie in ∂H k for k = i, j or not. Then F ij becomes a disc. Since P gives the simple proper handlebody decomposition H, the boundary ∂F ij has either at least two vertices or no vertex of P.  Suppose F ij is a disc and ∂F ij has more than two vertices. Let β be a sub-arc of ∂F ij cut off by the vertices, γ a properly embedded arc in F ij parallel to β, and H k (k = 1, 2, or j < k) the handlebody with β ⊂ ∂H k (figure 5a). We perform either a 2-0 move along β after a type-1 stabilization along γ or a 2-3 move along β according to whether there exists a different handlebody H l (l = 1, 2 or j < l) from H k with ∂β ⊂ H k ∩ H l or not (figure 5b,c). Each operation reduces the number of vertices in ∂F ij by two or one. By continuing this process, the number of vertices in ∂F ij can be reduced to two. Then we have F ij = ∅ after performing a 2-0 move on F ij .
Suppose F ij is a disc and ∂F ij has no vertices. There is a handlebody H k (k = 1, 2, or j < k) with ∂F ij ⊂ H k . In other words, F ik and F jk share their boundary components in ∂F ij . Since P is connected, at least one of F ik and F jk has another boundary component. If F ik shares another boundary component with F il (and F kl ), we take an arc in F ik which connects ∂F ij and ∂F il . Then we can remove F ij by a 2-0 move after applying a 0-2 move along the arc. A disc component of F jl arises in this operation. It follows that l > j or l = 1, 2, as (i, l) is greater than (i, j) in the lexicographical order by the minimality of (i, j). If F jk shares another boundary component with F jl (and F kl ), similarly, we can remove F ij by a 2-0 move after applying a 0-2 move. In this case, there is a possibility that F il changes to a disc from the empty set by this operation with 3 ≤ l < j. This implies that the minimal element of J varies from (i, j) to (i, l). In such a case, we take an oriented arc in P from a point in ∂F ij to a point in ∂H 1 or ∂H 2 . Then we can remove F ij , and the minimal element of J increases after successively applying 0-2 and 2-0 moves along the arc from the start to the end.
By repeating the same process, we have J = ∅. Namely, F ij = ∅ for 3 ≤ i < j ≤ n. Since each vertex of P is contained in four different handlebodies, this condition implies that P has no vertex.
Step 1. For each j ≥ 3, we will deform F 2j into a disc by applying similar operations in Step 0. Since ∂H j = F 1j ∪ F 2j , we may assume that F 2j is connected, if necessary, by performing type-1 stabilizations along arcs in F 1j . Take a maximal set of non-separating arcs properly embedded in F 2j . By performing type-1 stabilizations along the arcs, F 2j becomes a disc. Then H 2 ∪ · · · ∪ H n is a handlebody. By applying the same process for H , H 2 ∪ · · · ∪ H n becomes a handlebody. Hence (H 1 , (H 2 ∪ · · · ∪ H n )) and (H 1 , (H 2 ∪ · · · ∪ H n )) are Heegaard splittings of M. By the Reidemeister-Singer theorem, these two Heegaard splittings become equivalent after performing a finite sequence of type-0 stabilizations. In particular, we can assume H 1 = H 1 .
Step 2. Similarly to Step 1, for each j ≥ 3, we can deform F 1j into a disc by performing type-1 stabilizations along suitable arcs properly embedded in F 1j .
Proof of claim 3.6. According to the deformation of H 2 at this step, there exist mutually disjoint stabilizations Figure 6. A deformation at Step 2. (Online version in colour.) handlebody H i at the end of Step 1.) We can take mutually disjoint arcs α i1 , . . . , α ig i properly embedded in F 2i so that α ij ∩ D = α ij ∩ D ij , and α ij intersects D ij transversely in a single point. See figure 6. Let E ij be a disc corresponding to Then the assertion holds since ( Let S 1 denote the surfaces F 12 at this stage. Claim 3.6 implies that any one-handle of each handlebody H i (i ≥ 3) can be a local one-handle after a handle slide on S 1 .
Step 3. For handle slide of H 3 , we will cover H 1 along S 1 with H 3 by type-1 stabilizations. Take a maximal set of mutually non-parallel, non-boundary parallel arcs properly embedded in S 1 whose endpoints lie in ∂H 3 . We perform type-1 stabilizations along those arcs. The surface Since all spines of S 1 are covered by H 3 , H 3 becomes a local unknotted handlebody after performing handle slides by claim 3.6 (figure 7). Applying the same process for H and arranging genera of H 3 and H 3 by performing type-0 stabilizations if necessary, we can assume that H 3 = H 3 .
According to the deformation of H 3 at this step, there exists a separating disc D 3 in H 3 that cuts off a handlebody can be regarded as the previous H 3 at the end of Step 2.) Let S 3 be the surface ∂V 3 \ (D 3 ∪ F 13 ), which is a subsurface of F 23 and homeomorphic to S 1 .
Step i (4 ≤ i ≤ n). At the beginning of Step i, we may have S 3 , . . . , S i−1 as subsurfaces of ∂H 3 , . . . , ∂H n−1 , respectively, that are homeomorphic to S 1 ⊂ ∂H 1 , and (i − 3) annuli A 1i , A 3i , . . . , A (i−2)i ⊂ ∂H 2 between ∂F 2i and a component of Similar to Step 3, we will cover H i−1 along S 1 with H i by performing handle slides of H i . By a 0-2 move and a 2-0 move on A 1i , F 1i extends to A 1i , and an annulus F 3i arises. Continuing the same becomes the annulus A ji . By the same operation as Step 3 on S i−1 , S i−1 \ F (i−1)i becomes the union of n − 4 annuli including A (i−1)(i+1) , . . . , A (i−1)n . (In the case of n = 4, F 34 includes S 3 at Step 4.) By the same argument at Step 3, then, H i can be a local unknotted handlebody after handle slide, and we can assume that H i = H i .
When we finish Step n, we have H 1 = H 1 , H 3 = H 3 , . . . , H n = H n . Since this automatically implies that H 2 = H 2 , the proof is completed.

Handlebody decomposition consisting of three handlebodies
In this section, we provide several results of handlebody decompositions consisting of three handlebodies. We keep assuming that all handlebody decompositions are simple and proper unless otherwise specified.
Section 4a will consider stabilizability on handlebody decompositions containing a three-ball. In [12], Waldhausen showed that any genus-g Heegaard splitting of S 3 is stabilized for g ≥ 1.
On the other hand, Koenig found an infinite family of unstabilized type-(1, 2, 2) handlebody decompositions of S 3 (see §6 in [5]). We will show that a closed connected orientable threemanifold not containing a non-separating sphere admits an unstabilized type-(0, 0, g) handlebody decomposition, where g is the Heegaard genus of the manifold (proposition 4.2). Furthermore, we will see that almost all lens spaces admit a type-(0, 1, 2) handlebody decomposition (proposition 4.9). In §4b, we will study handlebody decompositions of the three-dimensional torus T 3 . These decompositions play an important role in polycontinuous patterns ( §5).

(a) Handlebody decompositions containing a three-ball
We first introduce the result of Gómez-Larrañaga [3]. That result gave a complete classification of all closed connected three-manifolds that admit handlebody decompositions with small genera.

Theorem 4.1 ([3, propositions 1-3, theorem 1]).
Let (H 1 , H 2 , H 3 ) be a type-(g 1 , g 2 , g 3 ) handlebody decomposition of a closed connected orientable three-manifold M with g 1 ≤ g 2 ≤ g 3 . We denote by B the connected sum of some copies of S 2 × S 1 , and denote by L or L i a lens space with non-trivial finite fundamental group. Then the following hold: (1) If all g i are equal to 0, then M is homeomorphic to S 3 or B. Conversely, S 3 and B admit such a handlebody decomposition. (2) If g 1 = g 2 = 0 and g 3 = 1, then M is homeomorphic to S 3 , B, L or B # L. Conversely, these manifolds admit such a handlebody decomposition.
Conversely, these manifolds admit such a handlebody decomposition.
denotes a Seifert fibre space with at most three exceptional fibres. Conversely, these manifolds admit such a handlebody decomposition.
Let M be a closed orientable three-manifold with a Heegaard splitting (W 1 , W 2 ) of genus l. Then, we can take l + 1 non-separating discs in W 1 so that they separate W 1 into two three-balls. Hence, M admits a type-(0, 0, l) handlebody decomposition (see [13, example 1.2]). The following proposition classifies such a decomposition. To prove the above proposition, we first show the following lemma. Proof. Let F ij denote a surface as in remark 2.2. We show that the surface F 12 consists of discs. Assume that F 12 contains a non-disc component S. Then there exists an essential simple loop C in S such that each complementary region of C in ∂H 1 ∼ = S 2 contains a connected component of F 13 . Since H 1 and H 2 are three-balls, the simple loop C bounds a disc in each of H 1 and H 2 . Then the union of the two discs is a non-separating disc, which is a contradiction.
Thus, F 12 consists of only discs. It follows that the union of H 1 and H 2 is a handlebody, which implies the assertion.
Proof of proposition 4.2. Let g be the Heegaard genus of a closed, connected, orientable threemanifold M. Then, as explained above, M admits a type-(0, 0, g) handlebody decomposition. Then, by remark 3.2, we can obtain a type-(0, 0, l) handlebody decomposition of M for each g ≤ l.
Conversely, assume that M admits a type-(0, 0, l) handlebody decomposition. By lemma 4.3, this handlebody decomposition induces a Heegaard splitting of genus l. Thus, we have g ≤ l. This particularly implies that a type-(0, 0, g) handlebody decomposition of M is unstabilized.
Example 4.4. An unstabilized type-(0, 0, 2) handlebody decomposition of S 3 is constructed as follows. Let (W 1 , W 2 ) be a genus-2 Heegaard splitting of S 3 . By using [14, §5 and fig. 4], we can take a non-primitive disc triple of W 1 , which separates W 1 into two three-balls. We denote the three-balls by H 1 and H 2 , and put H 3 = W 2 . Then (H 1 , H 2 , H 3 ) forms a type-(0, 0, 2) handlebody decomposition of S 3 . Because each component of F 12 is a non-primitive disc in W 1 , we can see that the boundary of any properly embedded disc in H 3 transversely intersects the singular graph of the partition in at least six points. Hence we cannot perform a destabilization along any properly embedded discs in H 3 . Therefore, the decomposition is unstabilized.
Next, we will consider the stabilizability of type-(0, 1, l) handlebody decompositions. Proof. We first assume that F 12 consists of discs. Then there exists a meridian disc of H 2 whose boundary intersects ∂F 12 transversely exactly twice. Hence we can perform a type-1 destabilization along the meridian disc.
In the remainder, we assume that F 12 has a non-disc component. Proof of claim 4.6. Suppose that χ (S) < 0. Since S ⊂ ∂H 1 ∼ = S 2 , the boundary ∂S has at least three components. Then there exists a component c of ∂S such that it is an inessential loop in ∂H 2 ∼ = T 2 . Hence c bounds a properly embedded disc in H 2 . The closed curve c also bounds a properly embedded disc in H 1 since H 1 is a three-ball. Because each complementary region of c in ∂H 1 ∼ = S 2 contains a component of F 13 , the two properly embedded discs in H 1 and H 2 form a nonseparating sphere. This contradicts the irreducibility of M. Proof of claim 4.7. Assume that a core curve C of an annulus component of F 12 is inessential in ∂H 2 . Then C bounds a properly embedded disc in H 2 , and each complementary region of C intersects F 23 . Since H 1 is a three-ball, C also bounds a properly embedded disc in H 1 . Hence the two discs form a non-separating sphere. This is a contradiction. Proof of claim 4.8. We assume that F 12 contains two annulus components. Then, by claim 4.7, their core curves, C 1 and C 2 , are parallel essential loops in ∂H 2 ∼ = T 2 . Thus, C 1 ∪ C 2 cobounds a properly embedded annulus in H 2 . Since H 1 is a three-ball, each of C 1 and C 2 bounds a disc in H 1 . Because each complementary region of C 1 ∪ C 2 in ∂H 2 intersects F 23 , the union of the annulus and the discs is a non-separating sphere. This is a contradiction.  Figure 8. An unstabilized type-(0, 1, 2) handlebody decomposition of a lens space with non-trivial finite fundamental group. The decomposition consists of three handlebodies Let C be a core curve of the annulus component A of F 12 . Then we have [C] = aμ + bλ ∈ H 1 (∂H 2 ), where a, b ∈ Z, and μ and λ denote homology classes of a meridian loop and a longitude loop, respectively. Since H 1 is a three-ball, each component of ∂A bounds a properly embedded disc in H 1 . Then the union S of ∂H 2 \ int(A) and the two discs is a separating sphere in M. Thus, if b = 0 and ±1, then M has a lens space as a connected summand. However, M is irreducible and not a lens space. Hence we have b = 0 or ±1. If b = 0, then C bounds a properly embedded disc in H 2 . The curve C also bounds a properly embedded disc in H 1 since H 1 is a three-ball. So, the discs form a non-separating sphere, which is a contradiction. Hence we have b = ±1. Thus, we can take a meridian disc of H 2 that intersects the boundary of F 12 transversely exactly two points. Therefore, we can perform a type-1 destabilization along the meridian disc.
The following proposition implies that the assumption that M is not a lens space in proposition 4.5 is essential. Proof. Let (W 1 , W 2 ) be a genus-2 Heegaard splitting of S 3 . Then there exists a pair of nonprimitive discs D 1 and D 2 in W 1 as in example 4.4. Note that B := N(D 1 ; W 1 ) ∪ N(D 2 ; W 1 ) ∪ W 2 is a three-ball, where N(D 1 ; W 1 ) and N(D 2 ; W 1 ) are regular neighbourhoods of D 1 and D 2 , respectively. We take an unknotted arc δ in the interior of W 1 that joins D 1 and D 2 and intersects them at only its endpoints. Let h be a one-handle attached to sides of each N(D 1 ; W 1 ) and N(D 2 ; W 1 ) along δ. Then, h ∪ B is a solid torus (figure 8). Hence, for any lens space M, there exists a homeomorphism ψ from ∂(h ∪ B) to the boundary of a solid torus H 2 such that M is homeomorphic to the manifold pasted by h ∪ B and H 2 along ψ. We put Thus, (H 1 , H 2 , H 3 ) is a type-(0, 1, 2) handlebody decomposition of M. By the construction, each meridian disc of H 2 and H 3 intersects the singular graph at least four and six times, respectively. Hence, the handlebody decomposition is unstabilized.   Theorem 4.1 guarantees that T 3 admits a type-(1, 1, 1) handlebody decomposition. Figure 9c shows a decomposition of T 2 into three hexagons. By taking the product with S 1 , we have a decomposition of T 3 into three solid tori. We call this handlebody decomposition the hexagonal honeycomb decomposition (figure 9d). In general, a three-manifold admits a lot of handlebody decompositions of the same type. The next proposition asserts that the hexagonal honeycomb decomposition is the unique type-(1, 1, 1) handlebody decomposition of T 3 up to self-homeomorphism of T 3 .
Proof of claim 4.12. Suppose there is a disc component D in some F ij . Without loss of generality, we can assume that D ⊂ F 23 . If ∂D is essential in ∂H 1 , H 1 = H 1 ∪ N(D) is a punctured lens space. Since T 3 is prime, H 1 is a three-ball. Thus, a triple (H 1 , cl(H 2 \ N(D)), cl(H 3 \ N(D))) gives a simple and proper type-(0, 1, 1) handlebody decomposition of T 3 . However, from proposition 4.10, T 3 does not admit such a decomposition, which is a contradiction.
Suppose ∂D is inessential in ∂H 1 . Then we can take a disc D in ∂H 1 such that ∂D = ∂D. Put S = D ∪ D . Since T 3 does not contain non-separating spheres, S is separating. Thus, F 23 consists of only the disc D. Hence, H 2 ∪ H 3 is a genus-2 handlebody. This is impossible because ∂(H 2 ∪ H 3 ) = ∂H 1 is a torus.
By claim 4.12 and [3, lemma 1], each component of F ij is an annulus. Suppose the core of an annulus of F ij is meridional in H i . Then, C is longitudinal in H j ; otherwise, we can find a punctured lens space or a punctured S 2 × S 1 in T 3 , which is a contradiction. Then, by removing the neighbourhood of a meridian disc in H i and attaching it to H j , we have a decomposition of T 3 with two three-balls and a solid torus, i.e. a type-(0, 0, 1) decomposition, which contradicts proposition 4.10. Therefore, H 1 , H 2 and H 3 are fibre tori of a Seifert fibration of T 3 . As the Seifert fibre structure of T 3 is unique up to self-homeomorphism of T 3 , we can assume that 1, 2, 3), where D 1 , D 2 and D 3 are discs in T 2 satisfying D 1 ∪ D 2 ∪ D 3 = T 2 . By the Euler characteristic, the intersection of two of the discs consists of precisely three arcs. Hence, this structure corresponds to the hexagonal honeycomb decomposition in T 3 . Figure 10. (b,d) A pair of different handlebody decompositions of T 3 . The decomposition (b) (resp. (d)) is of type-(2, 2, 2) obtained from the hexagonal honeycomb decomposition by type-1 stabilizations along α 1 , α 2 and α 3 in (a) (resp. β 1 , β 2 and β 3 in (c)). (Online version in colour.) When it comes to the case of type-(2, 2, 2) handlebody decompositions, the uniqueness no longer holds, as we see in the following example.
Example 4.13. By performing type-1 stabilizations to the honeycomb decomposition three times, we have the two type-(2, 2, 2) handlebody decompositions shown in figure 10. In figure 10a, each F ij = H i ∩ H j is homeomorphic to the disjoint union of a two-holed torus and a disc. On the other hand, in figure 10b, the surface F 23 is homeomorphic to the disjoint union of a one-holed torus and an annulus, and each F 1j is homeomorphic to a three-holed sphere. Hence, they are different decompositions.

Topological study of polycontinuous patterns
In this section, we will consider 'polycontinuous patterns', which are roughly three-periodic structures assembled by polymers. See, for example, [6,15] for studies on polycontinuous patterns. We will suggest a mathematical model of polycontinuous patterns (definition 5.8).

(a) Polycontinuous patterns and net-like patterns
First, we define 'net-like patterns' that satisfy the essential properties of polycontinuous patterns. Definition 5.1. We denote by T d the d-dimensional torus. Let X be a graph embedded in R d such that each component of X is unbounded. If there exists a covering map π : R d → T d such that all covering transformations of π preserve X, then X is called a net.
In this paper, we mainly discuss the case where d = 3.

Remark 5.2.
In crystal chemistry (e.g. [16]), the term 'net' means a periodic, connected, simple, abstract graph. In this paper, we allow a net to be disconnected. Furthermore, all nets are embedded in Euclidean space. Definition 5.3. Let P be a non-compact connected two-dimensional polyhedron embedded in R 3 . The polyhedron P is called a net-like pattern if there exist a covering map π : R 3 → T 3 and a net X such that the following conditions hold: (1) All covering transformations of π preserve both P and X.
(2) The polyhedron P divides R 3 into unbounded open components V i (i ∈ I), where I is a finite or countable set. (3) There exists a strong deformation retraction of R 3 \ P onto X.
We call the pair ( P, π ) a framed net-like pattern, and π its frame. We say that a connected component of R 3 \ P (resp. X) is a labyrinthine domain (resp. labyrinthine net) of P. A net-like pattern P is said to be proper if there is no simple closed curve in R 3 that does not cross the singular graph and intersects a sector of P transversely once. A net-like pattern P is said to be simple if P is a simple polyhedron.
More generally, we can define net-like patterns for any closed prime three-manifold with a (possibly non-Euclidean) crystallographic group and its covering space, but this paper will not deal with it.
Remark 5.4. Consider two net-like patterns that satisfy the following conditions: (1) They have the same labyrinthine net.
(2) They do not have a disc sector.
(3) The singular graphs of them have no vertices.
Then, they can be transformed to each other by a (possibly infinite) sequence of IX-moves, XI-moves and isotopies (see [17, theorem 3.1]).
By using [11,18], if two net-like patterns with the same labyrinthine net are simple, then the patterns can be transformed to each other by a (possibly infinite) sequence of 0-2 moves, 2-0 moves, 2-3 moves, 3-2 moves and isotopies.
The following two propositions state a relationship between (framed) net-like patterns and handlebody decompositions of T 3 . (H 1 , H 2 , . . . , H n ; P) be a handlebody decomposition of T 3 , and P the preimage of P under the universal covering map π of T 3 . Suppose that, for each i, the induced homomorphism (ι i ) * :

(T 3 ) is not trivial, where ι i is the inclusion map. Then the pair ( P, π ) is a framed net-like pattern. Furthermore, if P is simple (resp. proper), then the net-like pattern is also simple (resp. proper).
Proof. Let {V i j } be the connected components of the preimage of H i under π . Since the homomorphism (ι i ) * is not trivial, each open component V i j is unbounded. Each open handlebody H i contains a simple finite graph X i that is a strong deformation retract of H i . Then the preimage, X i , of X i under π is a net. Furthermore, each connected component of X i is a strong deformation retract of some V i j . Hence, ( P, π ) is a framed net-like pattern. Since π is a local homeomorphism, if P is simple, then the preimage P is also simple. Next, assume that the handlebody decomposition (H 1 , H 2 , . . . , H n ; P) of T 3 is proper, whereas the netlike pattern P is not proper. Then, there exists a simple loop c in R 3 that transversely intersects P at a single point only in a sector. Thus, there exists a simple loop in T 3 isotopic to π ( c) that intersects a sector of P transversely once. Hence, the handlebody decomposition is not proper, which is a contradiction. Therefore, the net-like pattern P is proper. Proposition 5.6. Let ( P, π ) be a framed net-like pattern.
(1) The image π ( P) gives a handlebody decomposition of T 3 . If P is simple, then the handlebody decomposition is also simple. (2) Let {V i } i∈I be the set of labyrinthine domains of P, where I is a finite or countable set. Suppose that P is proper. Suppose further that for any V i , V j with V i = V j , where V i is the image of V j under some covering transformation, V i and V j are not adjacent to the same sector. Then the handlebody decomposition given by π ( P) is proper.
Proof. Let Γ be the covering transformation group of π . Set P = π ( P).
(1) Since P is a connected two-dimensional polyhedron, its projection image P is also a connected two-dimensional polyhedron. Furthermore, if P is simple, then P is also simple.  since P is a net-like pattern, there exists a labyrinthine net X i such that X i is a strong deformation retract of V i . Put G = π ( X i ). Then, G is an embedding of a graph in T 3 . So, the fundamental group π 1 (G) is free. Since π | V i is a covering map, and X i is a strong deformation retract of V i , the inclusion map G → H j induces an isomorphism from π 1 (G) to π 1 (H j ). Hence, H j is the interior of a handlebody because π 1 (H j ) is free. Therefore, P gives a handlebody decomposition of T 3 .
(2) We suppose that P is proper and P is not proper. Then there exists a simple loop c : [0, 1] → T 3 such that it transversely intersects P at a single point only in a sector. Let c be a lift of c. Since P is proper, c is an arc (not a loop) whose initial point v and terminal point w are contained in different labyrinthine domains V i and V j , respectively, of R 3 \ P. Note that V i and V j are adjacent. This is impossible because there exists a covering transformation that takes v to w, so V i to V j . Example 5.7. Figure 11a illustrates a simple proper net-like pattern that comes from the hexagonal honeycomb tessellation of R 2 . A yellow polygon illustrates a fundamental domain of its frame. We call the pattern the hexagonal honeycomb pattern. By proposition 5.6, the hexagonal honeycomb pattern with the frame induces the hexagonal honeycomb decomposition (see figure 9d). Note that a tessellation of R 2 induces a net-like pattern in general. The meanings of colours except yellow in figure 11a will be explained in definition 5.10.
We now suggest a strict mathematical definition of polycontinuous patterns. Definition 5.8. We say that a net-like pattern P is an n-continuous pattern (or a polycontinuous pattern) if the following conditions hold: (i) P has precisely n labyrinthine domains. (ii) P is proper. (iii) Any sector of P is not a disc.
Note that for any positive integer n, there exists an n-continuous pattern. In the remainder, we call a two-continuous (resp. three-continuous) pattern a bicontinuous (resp. tricontinuous) pattern, according to the conventions of soft materials [6,15].
The following corollary is a polycontinuous pattern version of proposition 5.5.
Corollary 5.9. Let (H 1 , . . . , H n ; P) be a proper handlebody decomposition of T 3 , and P the preimage of P under the universal covering map of T 3 . Suppose that the following two conditions hold: (2) Any sector of P is not a disc.
Then, P is a polycontinuous pattern. In particular, if (ι i ) * (π 1 (H i )) = π 1 (T 3 ) for each i, then P is an n-continuous pattern. Furthermore, if P is simple (resp. proper), then P is also simple (resp. proper).

(b) Colourings of patterns
In this subsection, we will define colourings of net-like patterns. Each labyrinthine domain of a net-like pattern is a mathematical model of polymers assembled in one kind of block. In materials science, one kind of block may form many domains of a net-like pattern in general. To describe such a situation, we introduce 'colours' of net-like patterns, of which each colour corresponds to one kind of block of polymers. Definition 5.10. Let P be a net-like pattern. Set X n = {1, 2, . . . , n}. Let V denote the set of all labyrinthine domains of P. A surjection ϕ : V → X n is called an n-colouring of P if there exists a frame π of P such that the following conditions hold: (1) For each covering transformation t of π and for any V ∈ V, we have ϕ(V) = ϕ • t(V).
(2) Two sides of a local part of each sector have different colours. Namely, for each point p in a sector C, the two labyrinthine domains, V and V , that have non-trivial intersection with N(p) satisfy ϕ(V) = ϕ(V ).
The image ϕ(V) is called the colour of V. The frame π of P is said to be compatible with the colouring ϕ. A net-like pattern together with a fixed (n-) colouring is called an (n-)coloured net-like pattern (figure 11b). We say that two coloured net-like patterns, P and Q, are equivalent if there exists an ambient isotopy of R 3 that moves P to Q, and each pair of corresponding labyrinthine domains has the same colour after permuting the colours. If a surjection ϕ satisfies only the condition (1), then we call ϕ a non-effective n-colouring, and P is said to be non-effectively n-coloured. Let P be a (possibly non-effectively) n-coloured net-like pattern with a colouring ϕ : V → X n , where X n = {1, 2, . . . , n}, and π a frame of P compatible with ϕ. By proposition 5.6, the image π ( P) gives a handlebody decomposition of T 3 . Denote by V the set of all handlebodies of the decomposition. Then we say that ( P, π ) is of type (g 1 , . . . , g n ), where g i = [g i 1 , . . . , g i k ] is a sequence of the genera of the handlebodies in V coloured by i ∈ X n . (For simplicity, if the length of g i is equal to 1, then we put g i = g i 1 .) Note that, as opposed to colourings of graphs on surfaces, for any integers m, n with n ≥ 2 and m < n, there is an n-coloured framed net-like pattern that does not admit m-colouring. Remark 5.11. If a net-like pattern admits a colouring, then it is necessarily proper.
In fact, a coloured net-like pattern P with its frame π compatible with the colouring satisfies the condition that π ( P) induces a proper handlebody decomposition of T 3 since any two labyrinthine domains sharing a sector have different colours (see proposition 5.6). Hence, the following holds. k i ] be a sequence of positive integers for i ∈ {1, . . . , n}. Suppose that ( P, π ) admits an n-colouring and is of type (g 1 , . . . , g n ). Then π ( P) gives a proper type-(g The converse of the above corollary is clear by proposition 5.5. k n ) handlebody decomposition of T 3 and let π be the universal covering map R 3 → T 3 . We assume that H (i) We further assume that, for each handlebody H j is the inclusion map. Then (π −1 (P), π ) is a coloured net-like pattern of type (g 1 , . . . , g n  (c) A sufficient condition for the equivalence of patterns Corollaries 5.12 and 5.13 say there is a nice relationship between coloured net-like patterns and proper handlebody decompositions. This subsection gives a sufficient condition for two coloured net-like patterns to be equivalent.
To this end, we first consider adjusting a framed net-like pattern to another frame. Let ( P, π ) be a framed net-like pattern, and let ρ be a covering map R 3 → T 3 . Since the two covering maps are equivalent, there exists a self-homeomorphism f of R 3 such that π = ρ • f . If f is orientationpreserving, we say that π and ρ have the same orientation. Otherwise, we say that π and ρ have different orientations.
If π and ρ have the same orientation, then P is isotopic to f ( P), and (f ( P), ρ) is a framed net-like pattern. Consider the case that π and ρ have different orientations. Let r be an orientationreversing self-homeomorphism of T 3 . Then, π := r • π is also a covering map R 3 → T 3 . Hence, there exists an orientation-preserving homeomorphism g : R 3 → R 3 such that π = ρ • g. So, P is isotopic to g( P), and we have r(π ( P)) = ρ(g( P)). Furthermore, (g( P), ρ) is a framed net-like pattern because each covering transformation of π is also that of π .
To summarize, we obtain the following lemma.
Lemma 5.14. Let ( P, π ) be a framed net-like pattern, and let ρ be a covering map R 3 → T 3 . Then, there exists a net-like pattern Q such that the following three conditions hold: (1) The covering map ρ is a frame of Q. (2) The pattern P is isotopic to Q.
In particular, if P is (non-effectively) n-coloured, then so is Q. Furthermore, P and Q have the same type.
By lemma 5.14, we can assume that any two net-like patterns have the same frame. Proposition 5.6 says the two framed net-like patterns induce two handlebody decompositions of T 3 , respectively. If the two decompositions are isotopic, then the two patterns are also isotopic. In fact, we can say more as follows.
Lemma 5.15. Let ( P, π ) and ( Q, π ) be framed net-like patterns. Suppose that there exists an orientationpreserving self-homeomorphism f of T 3 that maps π ( P) to π ( Q). Then, P is isotopic to Q.
Proof. By the assumption, we have an orientation-preserving self-homeomorphism f of T 3 that maps π ( P) to π ( Q). Let f be the unique lift of f • π . Then f is a homeomorphism of R 3 , and we have f ( P) = Q. Therefore, P and Q are isotopic.
By the above lemmas, we have the following proposition. Proposition 5.16. Let ( P, π ) and ( Q, ρ) be n-coloured framed net-like patterns. We assume that π ( P) and ρ( Q) are homeomorphic under an orientation-preserving or orientation-reversing self-homeomorphism f of T 3 according to whether the covering maps π and ρ have the same orientation or different orientations. Suppose that any two corresponding handlebodies under f are the images of labyrinthine domains with the same colour (after permuting the colours). Then ( P, π ) and ( Q, ρ) are equivalent.

Stabilizations on net-like patterns
In this section, we will discuss (de)stabilizations of net-like patterns and introduce some examples.  To do so, we define (de)stabilizations of net-like patterns. First, we will use an example to explain how to define it. In example 5.7, we introduced the hexagonal honeycomb pattern that is a three-coloured framed net-like pattern of type (1, 1, 1). We consider a type-1 stabilization on the pattern. Let α be a properly embedded arc in a sector of the pattern (figure 12a). We assume that α is lifted by π , i.e. the restriction of π to α is injective, where π is the frame of the pattern. By corollary 5.12, π ( P) gives a simple proper handlebody decomposition. As noted in example 5.7, P := π ( P) is a simple proper type-(1, 1, 1) handlebody decomposition (figure 9d). Since α is lifted and contained in a sector, the image α := π ( α) is a properly embedded arc in a sector of P. Furthermore, α connects two labyrinthine domains mapped to the same handlebody by π . So, the arc α connects the same handlebody. Thus, we can perform a type-1 stabilization along α. Hence, a type-(1, 1, 2) handlebody decomposition P is obtained by performing a type-1 stabilization along α. By corollary 5.13, the preimage of P under π gives a three-coloured framed net-like pattern P of type (1, 1, 2) (figure 12b). Hence, we obtain the new net-like pattern P from P. We will call such an operation a type-1 stabilization for net-like patterns.
Based on the above example, we give a strict definition of stabilizations for patterns as follows.
Definition 6.1. Let ( P, π ) be a simple, n-coloured, framed net-like pattern of type (g 1 , . . . , g n ), where g i is a sequence of positive integers [g Depending on the type of stabilization, we take an arc α as follows.
(type-0) The arc α is a properly embedded lifted arc in V that connects U and U . We assume that a lifted disc in V contains α as a part of its boundary, and the other part is contained in ∂V. (type-1) The arc α is a properly embedded lifted arc in a sector of P that connects V and V .
(type-2) The arc α is a properly embedded lifted arc in V that connects U and W. We assume that a lifted disc in V contains α as a part of its boundary, and the other part is contained in ∂V.
Then, we can obtain a new handlebody decomposition H performed by a suitable stabilization on H along π ( α). We can see by corollary 5.13 that the preimage of the partition of H is a simple, coloured, framed net-like pattern of type (g 1 , . . . , g n ). Here, each g i is equal to g i except for the following sequences: (type-1 and type-2) We call this operation a type-k stabilization (along α with respect to π ) and its inverse operation a type-k destabilization for each k. In electronic supplementary material, we discuss sufficient conditions for performing a destabilization.
Note that the result of a (de)stabilization of a polycontinuous pattern is not necessarily a polycontinuous pattern. Further note that in a type-2 stabilization along an arc for net-like patterns, even if the arc connects different labyrinthine domains, we cannot perform the operation if they are the same colour.
Definition 3.4 introduced some operations for handlebody decompositions called moves. We next consider a net-like pattern version of them. Of course, we can perform the original operations on simple coloured net-like patterns, but they generally lose periodicity after performing them. Thus, we give adapted 'moves' to net-like patterns in a similar way to the stabilizations. Definition 6.2. Let ( P, π ) be a simple, n-coloured, framed net-like pattern. Take a properly embedded lifted arc α in a sector (resp. an edge α of the singular graph of P) so that it connects labyrinthine domains V and V of different colours. By corollary 5.12, P := π ( P) gives a simple proper handlebody decomposition H. Then, we can obtain a new handlebody decomposition H performed by a 0-2 (resp. 2-3) move on P along π ( α). By corollary 5.13 the preimage of the partition of H is a simple, coloured, framed net-like pattern. We call such an operation a 0-2 (resp. 2-3) move (along α with respect to π ) and its inverse operation a 2-0 (resp. 3-2) move.
Note that, similar to type-2 stabilizations of net-like patterns, even if we can perform a move on a handlebody decomposition corresponding to a pattern, it does not necessarily mean that we can perform the corresponding move on the pattern.
An analogue of the Reidemeister-Singer theorem for net-like patterns is as follows.
Proof. We assume that π and ρ have the same orientation. The proof of the other case is similar. By corollary 5.12, the images P := π ( P) and Q := ρ( Q) give type-(g 1 , . . . , g n ) and type-(g 1 , . . . , g n ) simple proper handlebody decompositions of T 3 , respectively. Hence, by theorem 3.5, there exists a simple proper handlebody decomposition such that π ( P) and ρ( Q) are isotopic to the partition R of the decomposition after applying 0-2, 2-0 and 2-3 moves, and type-0 and type-1 stabilizations to them finitely many times. By corollary 5.13, R := π −1 (R) is a simple n-coloured net-like pattern. Therefore, by proposition 5.16, each of P and Q is equivalent to R after applying 0-2, 2-0 and 2-3 moves, and type-0 and type-1 stabilizations finitely many times.
In the above corollary, we assume that, for each colour, all labyrinthine domains coloured by it are mapped to the same handlebody because moves performed in the proof of theorem 3.5 generally do not preserve the colouring. On the concept of colourings, we can regard singlecoloured domains as composed of the same kind of blocks, so connecting these parts is a natural idea.  Let ( P, π ) be a simple, n-coloured, framed net-like pattern. Take a properly embedded lifted arc α in a sector so that it connects labyrinthine domains, V and V , of the same colour. We assume that H := π (V) and H := π (V ) are different handlebodies of the simple proper handlebody decomposition induced by P := π ( P). By performing a 0-2 move on P along π ( α), the modified H and H are intersected, and by corollary 5.12, their intersection consists of only the disc created by the operation. So, H := H ∪ H is also a handlebody. Hence, we have a new handlebody decomposition by replacing H and H with H . By corollary 5.13, the decomposition induces a new simple, coloured, framed net-like pattern ( P , π ). We call such an operation a domain-connection (along α with respect to π ) and its inverse operation a domain-disconnection. Remark 6.5. We can obtain the type of ( P , π ) in definition 6.4 as follows. Let g i be the sequence of positive integers in the type of ( P, π ) corresponding to the colour of the labyrinthine domains V and V . We remove the integers corresponding to V and V from g i and append their sum. Then, we denote a new sequence by g i . By replacing g i with g i , we obtain the type of ( P , π ).
By applying the following to a coloured net-like pattern, it satisfies the assumption of corollary 6.3. Lemma 6.6. Let ( P, π ) be a simple, n-coloured, framed net-like pattern of type (g 1 , . . . , g n ), where g i is a sequence of positive integers [g k . Then, we have a simple, ncoloured, framed net-like pattern of type ( g 1 , . . . , g n ) by applying 0-2 moves and domain-connections with respect to π finitely many times to P.
Proof. Since P is connected, for each colour i, there exist labyrinthine domains, V and V , with colour i and an embedded lifted arc δ joining V and V in P such that the following hold: (1) The images π (V) and π (V ) are distinct handlebodies.
(2) The arc δ does not cross any labyrinthine domains with colour i.
(3) The arc δ intersects the singular graph of P transversely.
Then, by cutting δ at its intersection with the singular graph, we obtain the sequence of sub-arcs δ 1 , . . . , δ k . Thus, we can perform 0-2 moves along δ 1 , . . ., δ k−1 , and we can finally apply domainconnection along δ k . By repeating the above process, all labyrinthine domains with colour i are joined. Then, the type corresponding to colour i is g i by remark 6.5. Therefore, we have a net-like pattern of type ( g 1 , . . . , g n ).

(b) Microphase separation of a block copolymer melt
One motivation for this research comes from materials science. We are interested in the characterization of polycontinuous patterns that appear as microphase separation of a block copolymer melt [6,7].
In this subsection, we discuss block copolymers and phase separation of a block copolymer melt. One reference of this subject is [19]. A polymer is a molecule of high molecular weight created by chemically coupling large numbers of small reactive molecules, called monomers. If a polymer is made of one type of monomer, it is called a homopolymer. A polymer containing two or more chemically distinct monomers is referred to as a copolymer. A block copolymer is an important type of copolymer, in which monomers of a given type form polymerized sequences called blocks. If a block copolymer contains two (respectively three) blocks, it is called a diblock (resp. triblock) copolymer. If a linear diblock copolymer is made of blocks of monomers A and B, it is called an AB diblock copolymer. An ABA triblock copolymer is a linear triblock copolymer consisting of a sequence of a block of monomer A, a block of monomer B, and a block of monomer A. See  A block copolymer melt is a solvent-free viscoelastic liquid composed of block copolymers. Due to the chemical distinction of monomers, we can observe phase separation in a block copolymer melt. A domain of phase separation consists of monomers of one type. Microphase separation of a block copolymer melt is phase separation with domains of the mesoscopic size scale. Sphere, cylinder, bicontinuous and lamellar structures appear as microphase separation of AB diblock or ABA triblock copolymers [19,20].
An example of bicontinuous patterns is the Gyroid surface. In materials science, in the bicontinuous pattern of an AB diblock copolymer melt, the domain of the A monomer is the neighbourhood of the partition surface, and that of the B monomer forms two labyrinths (figure 13b). A tricontinuous pattern is a mathematical model of microphase separation of an ABC star-shaped block copolymer melt. The branch line of a tricontinuous pattern consists of the connection points of the A, B and C blocks in the block copolymers [6]. See [21][22][23] for studies on geometric phases of star polymer melts. Note that a sector of a tricontinuous pattern is the interface of two domains.
Next, we discuss a mathematical model of microphase separation with four phases. Let A, B, C and D be four chemically distinct monomers. We consider the polycontinuous pattern of melts of four types of three-star block copolymers of ABC, ABD, ACD and BCD. In this case, four different branched lines appear. The interface of domains and these branched lines form a simple polyhedron. The vertex of the simple polyhedron of the polycontinuous pattern corresponds to the point where four domains A, B, C and D meet. The four edges corresponding to the connecting points of the ABC, ABD, ACD and BCD triblock star polymers are placed around a vertex. Also, ABCD four-star polymers are synthesized [24,25], and their morphologies have been discussed in [26,27]. The joining point of four blocks of the block copolymer corresponds to a vertex of the simple polycontinuous pattern.
We want to analyse the property of materials with this structure via a topological study of these polycontinuous patterns. We hope the characterization and the classification of polycontinuous patterns will lead to the design of polymeric materials with the desired properties.
As an application of corollary 6.3, we can discuss the relation between two microphaseseparated structures of the same type. Here, we discuss the polymer science implications of stabilization and destabilization operation of patterns.
Observation 6.7. The type-0 destabilization for a bicontinuous pattern can be considered as the model of the cancelling of an unstable local one-handle structure of the pattern of the microphase separation. The type-1 destabilization (resp. stabilization) for a polycontinuous pattern can be considered as the model of the separation (resp. amalgamation) of the domains during the uniaxial elongation of polymeric materials.  Figure 14. (a) An srs net. The orange line l passes through the points (0, 0, 8) and (8,8,0). Note that this net is topologically the same as the srs-b net (see [30]). (b) A 3srs net. The 2π/3 rotation around l preserves the net. (Online version in colour.)

(c) Example: a 3srs pattern
A 3srs pattern is an example of a tricontinuous pattern. In this subsection, we will show the pattern can be destabilized to the hexagonal honeycomb pattern. First, we introduce a 3srs net. An srs net is a 3-periodic 'minimal' net in R 3 (see [28] and figure 14a). Figure 14a illustrates an srs net with a cubical fundamental domain, of which the length of each edge is 8. The net is an infinite trivalent graph, and the space group of it is I4 1 32 (see [15,29]). Note that a 2π/3 rotation around the cube diagonal (shown in figure 14a) generates an action of order 3 and preserves the cube. A 3srs net is the union of the images of the srs net under the action (figure 14b). Figure 15a illustrates a branched surface in R 3 with a cubical fundamental domain. The branched surface is the union of precisely three surfaces with the boundary (figure 15b-d). It is clear that the branched surface is a simple three-coloured tricontinuous pattern, and each component of the 3srs net is a labyrinthine net of the pattern. We call the tricontinuous pattern the 3srs pattern. The 3srs pattern is of type (5,5,5)  Proof. Let P be the 3srs pattern, and π its frame obtained from a cubic fundamental domain as shown in figure 15. Put P = π ( P). Figure 16 shows a simple proper type-(5, 5, 5) handlebody decomposition (H 1 , H 2 , H 3 ; P) of T 3 induced by P. We denote by F 12 , F 13 and F 23 surfaces with boundary as in remark 2.2. By definition 6.1, if we destabilize the decomposition to the hexagonal honeycomb decomposition by performing a finite sequence of type-1 destabilizations, then we can also destabilize P to the hexagonal honeycomb pattern by corresponding destabilizations.
First, for each i, we take three meridian discs D i1 , D i2 and D i3 of the handlebody H i as shown in figure 16a-c. Each disc intersects the singular graph of P transversely exactly two points.   Furthermore, any two different discs are disjoint. Hence, we can perform type-1 destabilizations along them. By this operation, we obtain a type-(2, 2, 2) handlebody decomposition of T 3 (see figure 17a). For simplicity, we denote each handlebody and the partition of the destabilized handlebody decomposition by the same symbol H 1 , H 2 , H 3 and P, respectively. Note that the preimage of the union of spines of H 1 , H 2 and H 3 is isotopic to a 3hcb net as shown in figure 17b. See [31] for examples of materials with this chemical framework. See also [32]. The destabilized net-like pattern is also a simple coloured tricontinuous pattern. For the type-(2, 2, 2) handlebody decomposition, we can perform a type-1 destabilization along a meridian disc D 4 of H 3 (figures 17a and 18a). The type of resulting decomposition is (2, 2, 1). Figure 18a-e illustrates a destabilization to the type-(2, 2, 1) handlebody decomposition, which produces a type-(1, 1, 1) handlebody decomposition. The type-(1, 1, 1) handlebody decomposition illustrated in figure 18f is the hexagonal honeycomb decomposition (figure 9d).

Characterization of patterns
In this section, we will prove that bicontinuous patterns are unique. We will also show that simple, coloured, framed net-like patterns of type (1, 1, 1) are unique. On the other hand, we will provide two different simple coloured net-like patterns of type (1, 1, 1, 1).

(a) Bicontinuous patterns and Heegaard splittings of T 3
By definition, an n-continuous pattern consists of precisely n labyrinthine domains, and it is proper. Hence, by assigning a different colour to each domain, the pattern admits an n-colouring. In general, a frame of the pattern is not compatible with the colouring. However, by expanding the fundamental domain, we can obtain a frame compatible with the colouring. Then, by corollary 5.12, the pattern with the frame gives a proper type-(g 1 , . . . , g n ) handlebody decomposition of T 3 . Hence, the pattern is a framed net-like pattern of type (g 1 , . . . , g n ). In particular, we note the following for each simple bicontinuous pattern and such a frame. Remark 7.1. Any simple bicontinuous pattern and its frame compatible with a colouring induce a Heegaard splitting of T 3 .
By [33,34], Heegaard splittings of T 3 are determined by their Heegaard genera. Hence, we can prove the uniqueness of bicontinuous patterns. Proof. Let ( P, π ) and ( P , π ) be bicontinuous patterns of types (g, g) and (g , g ), respectively. For the frame π , there exists a basis a 1 , a 2 , a 3 of R 3 such that the translations t i defined by the vectors a i generate the covering transformation group. We denote by T a group generated by translations t g −1 1 , t 2 and t 3 . Hence we have a covering map ρ : R 3 → R 3 /T ∼ = T 3 . Since the Euler characteristic of P/ρ is (g − 1)-times that of P/π , the surface P/ρ gives a Heegaard splitting of T 3 of genus (g − 1)(g − 1) + 1. Similarly, we can take a covering map ρ : R 3 → T 3 so that P /ρ also gives a Heegaard splitting of genus (g − 1)(g − 1) + 1. Therefore, by using [34, Théorème] and proposition 5.16, the two simple bicontinuous patterns are equivalent.
The Gyroid, the Schwartz D surface and the Schwartz P surface are famous triply periodic minimal surfaces that decompose R 3 into precisely two open components (see [35]), i.e. the surfaces are simple bicontinuous patterns. In [  the Gyroid to the Schwartz D surface and the Schwartz D surface to the Schwartz P surface by an explicit formula. Note that theorem 7.2 is a generalization of the result but does not give a formula for transformation between patterns.
(b) The uniqueness of framed patterns of type (1, 1, 1) We consider the hexagonal honeycomb pattern introduced in example 5.7. Recall that its pattern admits a colouring and a frame compatible with it, as in figure 11a. The pattern induces the hexagonal honeycomb decomposition of T 3 . Hence, the hexagonal honeycomb pattern with the frame is of type (1, 1, 1). By propositions 4.11 and 5.16, the hexagonal honeycomb pattern is a canonical model of simple coloured net-like patterns of type (1, 1, 1). Therefore, we have the following.
Theorem 7.3. Any simple, coloured, framed net-like pattern of type (1, 1, 1) is equivalent to the hexagonal honeycomb pattern.
Note that a simple three-coloured net-like pattern whose labyrinthine nets consist of lines is not necessarily equivalent to the hexagonal honeycomb pattern in general (see example 7.4). Also, there are distinct simple coloured net-like patterns of type (1, 1, 1, 1) (see example 7.5).
Example 7.4. We consider a tessellation of the plane R 2 by three kinds of tiles: square, hexagon and eight-sided polygon. Figure 19 shows a net-like pattern induced by the tessellation. The left side (figure 19a) illustrates a framed net-like pattern of type (1, 1, 1) that is not coloured since eightsided components are assigned to the same colour, and they are adjacent. On the other hand, the pattern admits a four-colouring (figure 19b). However, it is no longer type (1, 1, 1). This pattern is called [8, 6, 4; 8, 8, 6] in [23, fig. 8(k)] and the colouring given there corresponds to a coloured net-like pattern of type ( [1,1], [1,1], [1,1]). Example 7.5. Figure 20 illustrates two simple coloured net-like patterns, ( P a , ρ a ) and ( P b , ρ b ), of type (1, 1, 1, 1) with a cubical fundamental domain, where ρ a and ρ b denote their frames compatible with the colourings, respectively. We can see the two patterns are not equivalent as follows. Let X a and X b be nets associated with P a and P b . We consider the image (ι a ) * (π 1 (ρ a ( X a ))) and (ι b ) * (π 1 (ρ b ( X b ))), where ι a and ι b are the inclusion maps, respectively. By figure 20a (ι a ) * (π 1 (ρ a ( X a ))) is isomorphic to Z ⊕ Z. On the other hand, (ι b ) * (π 1 (ρ b ( X b ))) is isomorphic to Z ⊕ Z ⊕ Z by figure 20b. Hence, P a is not equivalent to P b . By theorem 7.3, any two simple coloured framed net-like patterns of type (1, 1, 1) are equivalent. However, simple coloured net-like patterns of type (1, 1, 1, 1) are not unique.
The labyrinthine nets of these types of patterns are called cubic rod (cylinder) packings [37] or weavings [38]. Many of those structures do not correspond to simple coloured net-like patterns.